# Solving Linear Equations

**1.5 Solving Linear Equations **

In this section we study various methods, algebraic and graphical , for solving

equations of the form

f(x) = g(x) (1)

where both f(x) and g(x) are linear functions .

To solve Equation (1) algebraically, isolate the variable on one side of the

equation (traditionally, the left side). This can be done thanks to the follow-

ing two properties of numbers .

**Property I.** Adding or subtracting the same number
to both sides of an

equation does not change the solution to the equation.

**Property II. **Multiplying or dividing both sides of an equation by a
nonzero

number does not change the solution to the equation.

**Remark 1**

The above two properties apply to any equation and not only for linear

equations.

**Example 1**

Solve the equation 1 - 2(3 - x) = 10 + 5x.

**Solution.**

Multiplying we find 1-6+2x = 10+5x or 2x-5 = 10+5x. Add -2x-10

to both sides to obtain 3x = -15. Divide both sides by 3 to obtain x = -5.

If Equation (1) involves fractions, it is usually
recommended to remove the

denominators by multiplying both sides of the equation by the least common

denominator.

**Example 2**

Solve.

**Solution.**

Multiply through by the least common denominator 12 to obtain 3(2x-3)+

4(5) = 2x+2(12). This can be rewritten as 6x+11 = 2x+24. Add -2x-11

to both sides to obtain 4x = 13. Divide both sides by 4 to otbtain x = 13/4

Now, Equation(1) can be solved geometrically. Indeed,
since both f(x) and

g(x) are linear functiones their graphs are straight lines. So we can enter

each function into a graphing calculator and graph to obtain two lines. If

the two lines are parallel then we say that Equation (1) has no solutions. If

the window shows only one line then we say that the equation has an infinite

number of solutions. If the window shows two intersecting lines then the

equation has a unique solution. This unique solution can be found by using

the key INTERSECT found in your calculator. (For the TI-83, this found

by pressing 2nd and CALC).

**Example 3**

Solve graohically the equation 1 - 2(3 - x) = 10 + 5x.

**Solution.**

Geometrically, the point (-5;-15) is the point of intersection of the two

lines f(x) = 1-2(3-x) = 2x-5 and g(x) = 10+5x as shown in the figure

below

**Solving Literal Equations **

By a literal equation we mean an equation involving many letters. For ex-

ample, in finance, an investment of P dollars for t years at a simple interest

rate of r grows to a balance given by the formula A = P(1 + rt).

Example 4**
**Solve the equation A = P(1 + rt) for t.

**Solution.**

Distributing on the right we find A = P + Prt. Subtract P from both sides

to obtain Prt = A - P. Now, divide both sides by Pr to obtain

We conclude this section with an application problem

**Example 5**

The value of a new computer equipment is $20,000 and the value drops at a

constant rate so that it is worth $ 0 after five years. Let V (t) be the value

of the computer equipment t years after the equipment is purchased.

(a) Find the slope m and the vertical intercept b.

(b) Find a formula for V (t).

(c) How long does it take for the computer's value to depreciate to $4000?

**Solution.**

(a) Since V (0) = 20; 000 and V (5) = 0 then the slope of V (t) is

and the vertical intercept is V (0) = 20; 000.

(b) A formula of V (t) is V (t) = -4; 000t + 20; 000.

(c) We must find t so that V (t) = 4000 or 20; 000 - 4; 000t = 4000. This

implies 4000t = 16; 000 or t = 4 years

Prev | Next |